On a \(100 \times 100\) board 100 rooks are placed that cannot capturing one another.
Prove that an equal number of rooks is placed in the upper right and lower left cells of \(50 \times 50\) squares.
On a plane there are 100 sheep-points and one wolf-point. In one move, the wolf moves by no more than 1, after which one of the sheep moves by a distance of no more than 1, after that the wolf again moves, etc. At any initial location of the points, will a wolf be able to catch one of the sheep?
Every evening Ross arrives at a random time to the bus stop. Two bus routes stop at this bus stop. One of the routes takes Ross home, and the other takes him to visit his friend Rachel. Ross is waiting for the first bus and depending on which bus arrives, he goes either home or to his friend’s house. After a while, Ross noticed that he is twice as likely to visit Rachel than to be at home. Based on this, Ross concludes that one of the buses runs twice as often as the other. Is he right? Can buses run at the same frequency when the condition of the task is met? (It is assumed that buses do not run randomly, but on a certain schedule).
\(2n\) diplomats sit around a round table. After a break the same \(2n\) diplomats sit around the same table, but this time in a different order.
Prove that there will always be two diplomats with the same number of people sitting between them, both before and after the break.
A gang contains 50 gangsters. The whole gang has never taken part in a raid together, but every possible pair of gangsters has taken part in a raid together exactly once. Prove that one of the gangsters has taken part in no less than 8 different raids.
On a plane, there are given 2004 points. The distances between every pair of points is noted. Prove that among these noted distances at least thirty numbers are different.
There are three sets of dominoes of different colours. How can you put the dominoes from all three sets into a chain (according to the rules of the game) so that every two neighbouring dominoes are of a different colour?
On every cell of a \(9 \times 9\) board there is a beetle. At the sound of a whistle, every beetle crawls onto one of the diagonally neighbouring cells. Note that, in some cells, there may be more than one beetle, and some cells will be unoccupied.
Prove that there will be at least 9 unoccupied cells.
Determine all solutions of the equation \((n + 2)! - (n + 1)! - n! = n^2 + n^4\) in natural numbers.
On the planet Tau Ceti, the landmass takes up more than half the surface area. Prove that the Tau Cetians can drill a hole through the centre of their planet that connects land to land.